A comprehensive characterization of the set of polynomial curves with rational rotation-minimizing frames

A rotation-minimizing frame $({\mathbf{f}}_1,{\mathbf{f}}_2,{\mathbf{f}}_3)$ on a space curve $\mathbf{r}(\xi )$ defines an orthonormal basis for ${\mathbb{R}}^3$ in which ${\mathbf{f}}_1={\mathbf{r}}^{\mathrm{\prime }}\mathrm{/}\mathrm{\mid }{\mathbf{r}}^{\mathrm{\prime }}\mathrm{\mid }$ is the curve tangent, and the normal-plane vectors ${\mathbf{f}}_2$, ${\mathbf{f}}_3$ exhibit no instantaneous rotation about ${\mathbf{f}}_1$. Polynomial curves that admit rational rotation-minimizing frames (or RRMF curves) form a subset of the Pythagorean-hodograph (PH) curves, specified by integrating the form ${\mathbf{r}}^{\mathrm{\prime }}(\xi )=\mathcal{A}(\xi )\mathbf{i}{\mathcal{A}}^{\ast }(\xi )$ for some quaternion polynomial $\mathcal{A}(\xi )$. By introducing the notion of rotation indicatrix and of core of the quaternion polynomial $\mathcal{A}(\xi )$, a comprehensive characterization of the complete space of RRMF curves is developed, that subsumes all previously known special cases. This novel characterization helps clarify the structure of the complete space of RRMF curves, distinguishes the spatial RRMF curves from trivial (planar) cases, and paves the way toward new construction algorithms.